Pictures would help. Many floor boards in old houses extend under the walls, and are not supported at the face of the wall, which makes replacement more difficult.

There is no "one size fits all" answer. A reputable floor contractor knowa best. If floor is level and well supported I would consider leaving it and installing new floor over it.

“Weak in areas” and ”uneven” and ”significant dip” make it sound like the floor boards could benefit from some additional support from beneath. ”Too thin to sand again” generally means the boards have already been sanded so that nails and tongue & groove are showing.…. the floor should still be able to be refinished/restored without.sanding again.